Road To String Theory

Rayyan Abdullah 

Ever since mankind crawled out of caves, curiosity and quest for understanding the nature and the need for making life better, drove us to the future marked with developments. From lighting the first fire to exploring the cosmos to development of most fundamental theories, like String Theory to describe the nature. 

Fundamental Foces

Through extensive research and experimentations by our predecessors, we have come to know, for now, that there are four fundamental forces through which matter interacts. Namely, strong, electromagnetic, weak and gravitational force. Since gravitational and electromagnetic forces have infinite range, we have been able to describe them classically. Newton described gravitation through his famous inverse square law, and a similar law for electric forces was given by Coulomb. Later, Einstein reformulated gravitational theory, still classically, by his General Theory of Relativity which is based on the idea, that the presence of mass bends space-time itself and as the name suggest, it is a generalized form of special theory of relativity where space and time are taken on the same footing. 

Electric and magnetic forces used to be thought of as different forces, but experiments by Hans Christian Oersted in 1819 showed that they are in fact inter related. Later developments in this regard by Andr´e-Marie Amp`ere, Michael Faraday and James Clerk Maxwell unified the two forces into one, called electromagnetic force. Special theory combined with electromagnetism leads us to classical electrodynamics, where field strength tensor incorporates both electric and magnetic fields and the corresponding equations of motions are called Maxwell’s equations.

Since strong and weak forces are short range forces, they cannot be described classically. Hence, the next major unification was performed by Abdus Salam, Steven Weinberg and Sheldon Glashow was a quantum one. Developed in the 20^th century by Erwin Schrodinger, Werner Heisenberg, Paul Dirac and others, the Quantum Mechanics which describes nature at small scales, was a paradigm shift. In quantum mechanics classical observables become operators which acts on the wave-function of a system and gives the corresponding observables. If two operators fail to commute, the corresponding observables cannot be measured simultaneously. But this was non relativistic theory, applicable for particles with velocity much smaller than the velocity of light (v << c). But for relativistic particles, we need to use relativistic quantum mechanics. But relativistic quantum mechanics gives us negative energy solutions and infinite probability densities. Hence its Hilbert space is incomplete. In order to get the complete Hilbert space, Classical Field theory was combined with Special Theory of Relativity to get Quantum Field Theory.

Strength of Fundamental Forces | Source: PhyiscsLibertexts

Standard Model & GUTs

Electromagnetic, strong and weak interactions were described quantum mechanically. Richard Feynman formulated quantum theory for electromagnetic force called quantum electrodynamics, these interactions are mediated by photons. Quantum theory for strong interactions is known as quantum chromodynamics whose mediating quanta is the gluon.  Electromagnetic and weak forces were unified through quantum theory by Salaam, Weinberg and Glashow. The unified is known as Glashow-Salam-Weinberg Model or Electroweak theory and the weak interactions are mediated by W and Z bosons. And they are all quantum field theories.

The electroweak theory together with QCD form the Standard Model of particle physics. Since these three forces are described by gauge theories, Standard Model is based on the gauge group SU(3) x SU(2) X U(1), where SU(3), SU(2) and U(1) are the gauge groups of strong, weak and electromagnetic interactions. Our current understanding of how the presently known particles (60 so far) and three of the fundamental forces are related to each other is encapsulated in the Standard Model of particle physics. Developed in the early 1970s, it has successfully explained almost all experimental results and precisely predicted a wide variety of phenomena. Over time and through many experiments, the Standard Model has become established as a well-tested physics theory. At currently achievable accelerator energies of 13 TeV, the standard model (SM) of Particle Physics  provides a successful and predictive theoretical description.

Standard Model of Elementary Particles | Source: CERN

Nevertheless, it is believed that the Standard Model is merely a low-energy effective description of a more fundamental theory. The reasons behind this are that, the Standard Model has many (about 20) free parameters (coupling constants, mixing angles, etc.) which have to be fixed by experiments. A fundamental theory of all interactions must not have free parameters, in fact it should give rise to fixed parameters.

Standard Model does not include the gravitational interactions. While this can be safely ignored at laboratory energies, there is nevertheless an energy range where gravity competes with the gauge interactions. The gravitational interaction becomes large at energies comparable with the Planck scale. It is in this regime that a quantum theory of gravity is needed. And any unified theory of all interactions must include the gravitational interaction.

The first attempts focused on improving on unification. They gave rise to the grand unified theories (GUTs). All interactions were collected in a simple group SU(5) in the beginning, but also SO(10), E6, and others [3]. It is believed that, at high energies, the three gauge interactions of the Standard Model comprising the electromagnetic, weak, and strong forces are merged into a single force. Although this unified force has not been directly observed, the many GUT models theorize its existence.

The meaning of unification is to have a consistent framework which includes both quantum theory and gravity. The predominant belief is that this mainly requires to ‘quantize gravity’, i.e., to reformulate Einstein’s theory of gravity as a quantum theory. It is quite plausible that the introduction of a dynamical space-time requires significant modifications of quantum theory as well.

String Theory

Considered the best candidate for unifying all four fundamental forces of nature, String theory is the most ambitious, exciting and challenging subject in modern theoretical physics. Developed in the 1960s, String theory is not, in contrast to general relativity and quantum field theory, a theory in the strict sense. There is, e.g., no axiomatic formulation and there is no set of defining equations of motion. Instead there is a set of rules which have been developed over the years. They have led to rather spectacular results and have passed all plausible consistency checks.

Idea of string theory began with Kaluza-Klein Theory. At the time when Einstein proposed his general relativity, with his idea of curved space-time, Theoder Kaluza and Oscar Klein wondered if there are more than 3 spatial dimensions. They proposed a 5^th dimension, which was spatial, such that it doesn't go on forever in space but it is so curved that space comes back on itself like a circle. They started from a higher-dimensional theory of gravity and compactified the theory to four dimensions, and ended up with four-dimensional gravity plus extra gauge interactions.  They effectively unified electromagnetism and gravity. Particles that have charge under electromagnetism would just be particles moving in the circular fifth dimension. The radius of this extra dimension would be related to the electric charge of particles. A charged particle would move in one direction in the fifth dimension, and an oppositely charged particle would move in the other. And as the momentum of two oppositely charge particles would cancel out when they collide, they would no longer move in the fifth dimension making their combined charge neutral. Although Kaluza and Klein's theory seemed to unify the forces of electromagnetism and gravity it wasn't widely accepted when it was published in 1921. It was mathematically true but the concept of extra spatial dimensions seemed very strange.

Small circular dimensions on a flat space

The Kaluza-Klein idea was later resurrected with String Theory. The basic characteristic of the theory is that its elementary constituents are extended strings rather than point-like particles as in quantum field theory and different particles corresponds to different oscillation modes of the strings.

Bosonic string theory, whose spectrum contains only bosons is only consistent in 26 dimensions. Clearly, this theory is cannot describe the real world, since it does not contain fermions. To include the fermions, we use supersymmetry, which is a symmetry (transformations) between bosons and fermions and consequently, every boson has a fermion super partner and vice versa. The theories that emerge after integrating bosonic string theory with supersymmetry are only consistent in 10 dimensions and they are known as superstring theories. I’m writing the word “theories”, because there are 5 of them, Type I, Type IIA & B, and two heterotic strings. They are connected through a web of dualities, which are transformations.

Compactifications

We live in a 4 dimensional space-time (3 space and 1 time dimensions). If superstring theories are to describe our world, then where are those 6 other dimensions? Why can’t we perceive them? Well they are believed to be compact, so small and curved that in order to detect them we need extremely high energies. Therefore, to describe the real world we compactify the remaining 6 dimensions. Compactification is usually done on Calabi Yau manifolds. 

Calabi-Yau Manifold

Conclusion

String theory remains a fascinating and complex field of study that has captured the imagination of scientists and the public alike. While it has yet to produce definitive experimental evidence, its mathematical elegance and potential for unifying fundamental physics continue to make it a subject of intense research and debate. As researchers continue to explore the theoretical and practical implications of string theory, it is certain to remain a topic of interest and discovery for years to come.